Properties

Label 490.2.c.e
Level $490$
Weight $2$
Character orbit 490.c
Analytic conductor $3.913$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 490.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.91266969904\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( \beta_{1} + \beta_{3} ) q^{3} - q^{4} + ( -1 - \beta_{1} + \beta_{2} ) q^{5} + ( -\beta_{1} + \beta_{3} ) q^{6} -\beta_{2} q^{8} -3 q^{9} +O(q^{10})\) \( q + \beta_{2} q^{2} + ( \beta_{1} + \beta_{3} ) q^{3} - q^{4} + ( -1 - \beta_{1} + \beta_{2} ) q^{5} + ( -\beta_{1} + \beta_{3} ) q^{6} -\beta_{2} q^{8} -3 q^{9} + ( -1 - \beta_{2} - \beta_{3} ) q^{10} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{11} + ( -\beta_{1} - \beta_{3} ) q^{12} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{13} + ( 3 - 2 \beta_{1} - 3 \beta_{2} ) q^{15} + q^{16} + 2 \beta_{2} q^{17} -3 \beta_{2} q^{18} + ( 4 - \beta_{1} + \beta_{3} ) q^{19} + ( 1 + \beta_{1} - \beta_{2} ) q^{20} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{22} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{23} + ( \beta_{1} - \beta_{3} ) q^{24} + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{25} + ( 2 + \beta_{1} - \beta_{3} ) q^{26} + ( -2 - 2 \beta_{1} + 2 \beta_{3} ) q^{29} + ( 3 + 3 \beta_{2} - 2 \beta_{3} ) q^{30} + ( -4 - 2 \beta_{1} + 2 \beta_{3} ) q^{31} + \beta_{2} q^{32} -12 \beta_{2} q^{33} -2 q^{34} + 3 q^{36} -2 \beta_{2} q^{37} + ( -\beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{38} + ( 6 + 2 \beta_{1} - 2 \beta_{3} ) q^{39} + ( 1 + \beta_{2} + \beta_{3} ) q^{40} + ( 6 - 2 \beta_{1} + 2 \beta_{3} ) q^{41} + ( -2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{43} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{44} + ( 3 + 3 \beta_{1} - 3 \beta_{2} ) q^{45} + ( 2 - 2 \beta_{1} + 2 \beta_{3} ) q^{46} + ( 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{47} + ( \beta_{1} + \beta_{3} ) q^{48} + ( -1 + 2 \beta_{1} + 2 \beta_{3} ) q^{50} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{51} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{52} + ( -2 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} ) q^{53} + ( 6 + 6 \beta_{2} - 4 \beta_{3} ) q^{55} + ( 4 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{57} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{58} + ( -4 + \beta_{1} - \beta_{3} ) q^{59} + ( -3 + 2 \beta_{1} + 3 \beta_{2} ) q^{60} + ( -6 + \beta_{1} - \beta_{3} ) q^{61} + ( -2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{62} - q^{64} + ( -1 + 2 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} ) q^{65} + 12 q^{66} + 8 \beta_{2} q^{67} -2 \beta_{2} q^{68} + ( -12 + 2 \beta_{1} - 2 \beta_{3} ) q^{69} + ( -6 + 2 \beta_{1} - 2 \beta_{3} ) q^{71} + 3 \beta_{2} q^{72} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{73} + 2 q^{74} + ( -\beta_{1} + 12 \beta_{2} + \beta_{3} ) q^{75} + ( -4 + \beta_{1} - \beta_{3} ) q^{76} + ( 2 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{78} + ( -2 - 2 \beta_{1} + 2 \beta_{3} ) q^{79} + ( -1 - \beta_{1} + \beta_{2} ) q^{80} -9 q^{81} + ( -2 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} ) q^{82} + ( \beta_{1} + \beta_{3} ) q^{83} + ( -2 - 2 \beta_{2} - 2 \beta_{3} ) q^{85} + ( -4 + 2 \beta_{1} - 2 \beta_{3} ) q^{86} + ( -2 \beta_{1} - 12 \beta_{2} - 2 \beta_{3} ) q^{87} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{88} -10 q^{89} + ( 3 + 3 \beta_{2} + 3 \beta_{3} ) q^{90} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{92} + ( -4 \beta_{1} - 12 \beta_{2} - 4 \beta_{3} ) q^{93} + ( -4 - 2 \beta_{1} + 2 \beta_{3} ) q^{94} + ( -1 - 4 \beta_{1} + 7 \beta_{2} - 2 \beta_{3} ) q^{95} + ( -\beta_{1} + \beta_{3} ) q^{96} + ( 4 \beta_{1} + 6 \beta_{2} + 4 \beta_{3} ) q^{97} + ( 6 \beta_{1} - 6 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{5} - 12 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{4} - 4 q^{5} - 12 q^{9} - 4 q^{10} + 12 q^{15} + 4 q^{16} + 16 q^{19} + 4 q^{20} + 8 q^{26} - 8 q^{29} + 12 q^{30} - 16 q^{31} - 8 q^{34} + 12 q^{36} + 24 q^{39} + 4 q^{40} + 24 q^{41} + 12 q^{45} + 8 q^{46} - 4 q^{50} + 24 q^{55} - 16 q^{59} - 12 q^{60} - 24 q^{61} - 4 q^{64} - 4 q^{65} + 48 q^{66} - 48 q^{69} - 24 q^{71} + 8 q^{74} - 16 q^{76} - 8 q^{79} - 4 q^{80} - 36 q^{81} - 8 q^{85} - 16 q^{86} - 40 q^{89} + 12 q^{90} - 16 q^{94} - 4 q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/3\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(3 \beta_{2}\)
\(\nu^{3}\)\(=\)\(3 \beta_{3}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/490\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\)
\(\chi(n)\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
1.22474 1.22474i
−1.22474 + 1.22474i
−1.22474 1.22474i
1.22474 + 1.22474i
1.00000i 2.44949i −1.00000 −2.22474 + 0.224745i −2.44949 0 1.00000i −3.00000 0.224745 + 2.22474i
99.2 1.00000i 2.44949i −1.00000 0.224745 2.22474i 2.44949 0 1.00000i −3.00000 −2.22474 0.224745i
99.3 1.00000i 2.44949i −1.00000 0.224745 + 2.22474i 2.44949 0 1.00000i −3.00000 −2.22474 + 0.224745i
99.4 1.00000i 2.44949i −1.00000 −2.22474 0.224745i −2.44949 0 1.00000i −3.00000 0.224745 2.22474i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 490.2.c.e 4
5.b even 2 1 inner 490.2.c.e 4
5.c odd 4 1 2450.2.a.bl 2
5.c odd 4 1 2450.2.a.bq 2
7.b odd 2 1 70.2.c.a 4
7.c even 3 2 490.2.i.f 8
7.d odd 6 2 490.2.i.c 8
21.c even 2 1 630.2.g.g 4
28.d even 2 1 560.2.g.e 4
35.c odd 2 1 70.2.c.a 4
35.f even 4 1 350.2.a.g 2
35.f even 4 1 350.2.a.h 2
35.i odd 6 2 490.2.i.c 8
35.j even 6 2 490.2.i.f 8
56.e even 2 1 2240.2.g.i 4
56.h odd 2 1 2240.2.g.j 4
84.h odd 2 1 5040.2.t.t 4
105.g even 2 1 630.2.g.g 4
105.k odd 4 1 3150.2.a.bs 2
105.k odd 4 1 3150.2.a.bt 2
140.c even 2 1 560.2.g.e 4
140.j odd 4 1 2800.2.a.bl 2
140.j odd 4 1 2800.2.a.bm 2
280.c odd 2 1 2240.2.g.j 4
280.n even 2 1 2240.2.g.i 4
420.o odd 2 1 5040.2.t.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.c.a 4 7.b odd 2 1
70.2.c.a 4 35.c odd 2 1
350.2.a.g 2 35.f even 4 1
350.2.a.h 2 35.f even 4 1
490.2.c.e 4 1.a even 1 1 trivial
490.2.c.e 4 5.b even 2 1 inner
490.2.i.c 8 7.d odd 6 2
490.2.i.c 8 35.i odd 6 2
490.2.i.f 8 7.c even 3 2
490.2.i.f 8 35.j even 6 2
560.2.g.e 4 28.d even 2 1
560.2.g.e 4 140.c even 2 1
630.2.g.g 4 21.c even 2 1
630.2.g.g 4 105.g even 2 1
2240.2.g.i 4 56.e even 2 1
2240.2.g.i 4 280.n even 2 1
2240.2.g.j 4 56.h odd 2 1
2240.2.g.j 4 280.c odd 2 1
2450.2.a.bl 2 5.c odd 4 1
2450.2.a.bq 2 5.c odd 4 1
2800.2.a.bl 2 140.j odd 4 1
2800.2.a.bm 2 140.j odd 4 1
3150.2.a.bs 2 105.k odd 4 1
3150.2.a.bt 2 105.k odd 4 1
5040.2.t.t 4 84.h odd 2 1
5040.2.t.t 4 420.o odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(490, [\chi])\):

\( T_{3}^{2} + 6 \)
\( T_{11}^{2} - 24 \)
\( T_{19}^{2} - 8 T_{19} + 10 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( ( 6 + T^{2} )^{2} \)
$5$ \( 25 + 20 T + 8 T^{2} + 4 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( -24 + T^{2} )^{2} \)
$13$ \( 4 + 20 T^{2} + T^{4} \)
$17$ \( ( 4 + T^{2} )^{2} \)
$19$ \( ( 10 - 8 T + T^{2} )^{2} \)
$23$ \( 400 + 56 T^{2} + T^{4} \)
$29$ \( ( -20 + 4 T + T^{2} )^{2} \)
$31$ \( ( -8 + 8 T + T^{2} )^{2} \)
$37$ \( ( 4 + T^{2} )^{2} \)
$41$ \( ( 12 - 12 T + T^{2} )^{2} \)
$43$ \( 64 + 80 T^{2} + T^{4} \)
$47$ \( 64 + 80 T^{2} + T^{4} \)
$53$ \( 144 + 120 T^{2} + T^{4} \)
$59$ \( ( 10 + 8 T + T^{2} )^{2} \)
$61$ \( ( 30 + 12 T + T^{2} )^{2} \)
$67$ \( ( 64 + T^{2} )^{2} \)
$71$ \( ( 12 + 12 T + T^{2} )^{2} \)
$73$ \( 400 + 56 T^{2} + T^{4} \)
$79$ \( ( -20 + 4 T + T^{2} )^{2} \)
$83$ \( ( 6 + T^{2} )^{2} \)
$89$ \( ( 10 + T )^{4} \)
$97$ \( 3600 + 264 T^{2} + T^{4} \)
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